Vol. 42, No. 1, 1972

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Some closure properties for torsion classes of abelian groups

Barry J. Gardner

Vol. 42 (1972), No. 1, 45–61

In an earlier paper (B. J. Gardner, Pacific J. Math., 33 (1970), 109-116) the torsion classes of abelian groups which are closed under pure subgroups were characterized, and §§3-6 of the present paper are devoted to generalizations of results appearing there. If 𝒞 is a homomorphically closed class of objects in an abelian category, a subobject A of an object B is called 𝒞-pure if it is a direct summand of every intermediate subobject X for which X∕A ∈𝒞. (This terminology is due to C. P. Walker). In particular, 𝒞 may be a torsion class. The following question is investigated: If 𝒯 and 𝒰 are torsion classes of abelian groups, when is 𝒯 closed under 𝒰-pure subgroups? Although ordinary purity is not 𝒰-purity for any torsion class 𝒰, a torsion class 𝒯 is closed under pure subgroups if and only if it is closed under 𝒯0-pure subgroups, where 𝒯0 is the class of all torsion groups.

In §5, for an arbitrary torsion theory (𝒰,𝒢) a rank function ( 𝒰-rank) is defined for nonzero groups in 𝒢. It is: shown that every torsion class closed under 𝒰-pure subgroups. is determined by its intersection with 𝒰 and the groups of 𝒰-rank 1 it contains. When 𝒰 = 𝒯0, the groups with 𝒰-rank 1 are the rational groups, so the earlier results for ordinary purity suggest that in general some refinement of the representation should be possible.

A further special case of the general problem is als α solved: Let X and Y be rational groups, T(X),T(Y ) the smallest torsion classes containing them. If X is a subring of the rationals then T(X) is always closed under T(Y )-pure subgroups; if not, the condition is satisfied if and only if X has a greater type than Y.

§7 is devoted to proving the following result: A torsion class is closed under countable direct products, i.e. direct products of countable sets of groups, if and only if it is determined by torsion-free groups.

Mathematical Subject Classification 2000
Primary: 20K10
Secondary: 18E40
Received: 20 October 1970
Published: 1 July 1972
Barry J. Gardner